A new technique--the analytic Green's function, effective Hamiltonian technique--is utilized in calculations of bound states and resonances at solid surfaces. The results for electronic states in compound semiconductors are found to be in excellent agreement with angle-resolved photoemission and other experimental measurements. We begin by using the analytic representation of the crystal Green's function in a study of phonons at the (100) surface of a face-centered cubic crystal. The previous study of this problem involved diagonalizing 63 x 63 matrices. Here the calculations are performed essentially analytically, with a computer used to evaluate closed-form expressions. It is found that there are two new types of singularities in the surface contribution to the density of states, which we call "zero-width antiresonances" and "extremal-point singularities." They arise from the singularities that are evident in the analytic representation of the crystal Green's function when some group velocity v(,3) approaches zero. Next we turn to the physically more interesting calculation of bound and resonant surface states at the relaxed (110) surface of 11 compound semiconductors--GaAs, GaP, GaSb, ZnSe, ZnTe, InAs, InP, InSb, AlAs, AlSb, and AlP. Because our technique is superior to previous techniques for locating resonances, we find considerably more resonant structure than has been reported in previous theoretical studies. For example, we obtain a new branch of resonances A(,2)' that were not found in any of several very detailed calculations for GaAs. ...
